Various embodiments of this disclosure relate to very-large-scale integration (VLSI) chip design and, more particularly, to outputting masks that print closely to desired layout patterns for chip designs.
VLSI chip design is a multi-billion dollar business. With technology node sizes shrinking to as low as fourteen nanometers, it can be difficult to get sufficient yield from an optical lithographic process.
Computational methods are often used to modify an input mask, which represents a desired circuit. A widely adopted technique for improving printability is inverse lithography, which consists of modifying the mask of a chip design by adding features to precompensate for expected distortion during imaging. Another technique is the creation of a Phase-Shifting Mask (PSM). This technique provides an additional degree of freedom, in the phase of the mask element.
An improved method for mask design has been proposed in the past, involving decomposing a polygonal representation of the desired mask into a set of edges and corners. The edges and corners are moved and nudged until the output pattern resulting from application of the mask meets certain criteria. This method has the drawback that changes are made locally to edges and corners and, as a result, an extra verification step is required to check for transfer of unwanted side-lobs in the resulting pattern.
Another proposed solution is Variational Edge Placement Error, an improvement of the original Edge Placement Error used by mask synthesis and inverse lithography tools to compute displacement of segments, i.e., small parts of the polygonal edges of the mask. A vertex-based table-lookup is used to compute an aerial image of the mask, by first decomposing the rectilinear polygons into a summation of rectangles, and then using the precomputed values of the rectangle imprints. These and similar shape decomposition techniques rely on large tables of precomputed values and generally allow only for edge movement, making it impossible to analytically determine sub-resolution assist feature (SRAF) placement.
Sampling both the mask and a filter (i.e., an aerial image) on a rectangular grid allows for implementation of discrete methods to solve the inverse lithography problem. It has been proven that adapted sampling is needed to avoid aliasing, which is inherent by the discontinuous nature of the rectilinear polygons, otherwise having an over- or underestimate of the Critical Dimension. It has also been demonstrated that approximating a continuous time convolution by a discrete convolution can be counteracted by sampling at least twice the Nyquist rate of the filter. Filtering is performed in the frequency domain by the use of Fast Fourier (FFT) transforms. These methods have a computational time that does not scale with the content of the input (which is the polygons' shape and number), and that time increases with the sampling frequency.
Use of FFT transforms assumes that the mask and the filter are periodic in space, which is often not the case. Thus, all methods based on discretization will be limited by these assumptions, and any compensation for the assumptions (e.g., oversampling) will not scale well in time. Avoiding the discretization can be performed by separating the polygons in a sum of geometric figures, edges, or vertices, but no fast methods have been proposed that take advantage of the precision offered.
In recent years, “pixelated” masks (i.e., sampled masks) have been widely discussed, as they offer flexibility in mask design. The problem has been reduced to an unconstrained, continuous function optimization, which has been solved by a steepest descent algorithm. While the possibility to independently set any pixel in the aerial image improves printability greatly, the resulting complexity of the mask makes it impossible to fulfill manufacturing constraints. In an attempt to lower complexity of the mask, a regularization framework is employed with complexity penalty terms. The complexity still remains high, but this correction introduces further errors in the printed mask pattern. This technique allows integration of sensitive concepts, such as partially coherent illumination, mechanical fluctuations of the optical imaging system, algorithm variability, focus variation, and thin-mask assumptions. While providing a fast method to obtain the discrete mask, the technique is still based on sampling, and therefore all results are affected by aliasing. Furthermore, the output of the algorithm consists of floating point values, which then have to be thresholded to create the mask. The entire algorithm depends on this nonlinear operation to obtain a manufacturable mask, but there exists no proof that the obtained mask is optimal, regardless of whether the corresponding floating point valued image was optimal.
Accordingly, previous methods for performing inverse lithography are either fast and imprecise, and thereby incompatible with mask design rules, or they are slow, impractical, or inflexible.